To give you an idea of how difficult that would be, think about the amount of energy a 200 pound astronaut moving at 17,000 miles per hour orbital speed has, and how to get rid of that. If my spit-ball math is correct, the energy in just the astronaut is enough to boil away 120 times his weight in water, starting from body temperature at 98.6 degrees F and ending as steam at 212 degrees F. So, that "egg" is going to need a lot of cooling, or a lot of rocket-powered deceleration before it hits the atmosphere, probably a combination.
The other part of this problem is G-forces, both from any deorbiting rocket and from the rest of the deceleration by atmospheric drag. That has to happen slowly enough to not break bones of suffocate the person because of inability to take a breath. Each astronaut has a "chair" made to fit his/her body. So, any "escape eggs" would probably need to be personalized for each crew member. And it would require some sort of control system to make sure that it didn't slam into the dense part of the atmosphere at too high a velocity and crush or vaporize the whole shebang.
Finally, if an egg can get an astronaut decelerated and through the atmosphere, it still needs to have some sort of parachute system for the final landing, because just a fall from a few hundred feet would still be fatal.
And, there needs to be some strong structure to hold the astronaut, cooling gear, parachute (and some other stuff not discussed) together for all of that trip down.
My guess is that it is more efficient to have one "egg" for several astronauts, rather than one egg for each. And, we already have such eggs, named "Dragon" and "Soyuz".
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